"3-manifolds and their invariants"의 두 판 사이의 차이
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imported>Pythagoras0 잔글 (찾아 바꾸기 – “<math>\frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2}\ln|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}|\,dt=\frac{2}{\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))=\frac{2}{\sqrt{7}}(Cl(2\pi /7})+Cl(4\pi/7})-Cl(6\pi/7}))</math>” �) |
imported>Pythagoras0 |
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| 1번째 줄: | 1번째 줄: | ||
| − | == | + | ==fundamental results on three manifolds== |
| + | * mostow-prasad rigidity | ||
| + | * geometrization | ||
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| − | + | ==two approaches== | |
| + | * using Kleinian groups | ||
| + | * using [[Triangulations and the Bloch group]] | ||
| + | |||
==maps between threefolds== | ==maps between threefolds== | ||
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* maps between aspherical 3 manifolds | * maps between aspherical 3 manifolds | ||
* aspherical threefolds = second and higher homotopy groups vanish | * aspherical threefolds = second and higher homotopy groups vanish | ||
| 50번째 줄: | 49번째 줄: | ||
| + | ==invariants== | ||
| + | * Turaev-Viro invariant (related to [[6j symbols (Racha coefficient)|6j symbols]]) | ||
| + | ** Kauffman and Line 'The Temperley Lie algebra recoupling theory and invariants of 3-manifolds" | ||
| + | ** Turaev-Viro "state sum invariants of 3-manifolds and quantum 6j-symbols) | ||
| + | * [[Chern-Simons invariant]] | ||
| + | * [[Kashaev's volume conjecture]] | ||
| + | * [[Triangulations and the Bloch group]] | ||
| + | * [[Volume of hyperbolic threefolds and L-values]] and volume of knot complements | ||
| + | * [[Number fields and threefolds]] | ||
2013년 2월 1일 (금) 05:43 판
fundamental results on three manifolds
- mostow-prasad rigidity
- geometrization
two approaches
- using Kleinian groups
- using Triangulations and the Bloch group
maps between threefolds
- maps between aspherical 3 manifolds
- aspherical threefolds = second and higher homotopy groups vanish
- JSJ decomposition http://en.wikipedia.org/wiki/JSJ_decomposition
- cutting M into
- Seifert fibered pieces ~ non hyperbolic pieces
- atoroidal pieces ~ hyperbolic pieces
- cutting M into
- Thurston's geometrization
- S^3, E\times S^2, Sol
- E^3, E\times H^2, SL_2
- H^3, Nil
Volume of knot complement
- KnotData[]
KnotData["FigureEight", "HyperbolicVolume"]
N[%, 20]
- Dedekind zeta funciton evaluated at 2 gives a number related to volume of 3-manifold
- Bloch-Wigner dilogarithm is involved
a problem
- Prove
$$ \begin{align} \frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2}\ln|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}|\,dt & =\frac{2}{\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7})) \\ & = \frac{2}{\sqrt{7}}(Cl(2\pi /7)+Cl(4\pi/7)-Cl(6\pi/7)) \end{align} $$ - a log tangent integral
invariants
- Turaev-Viro invariant (related to 6j symbols)
- Kauffman and Line 'The Temperley Lie algebra recoupling theory and invariants of 3-manifolds"
- Turaev-Viro "state sum invariants of 3-manifolds and quantum 6j-symbols)
- Chern-Simons invariant
- Kashaev's volume conjecture
- Triangulations and the Bloch group
- Volume of hyperbolic threefolds and L-values and volume of knot complements
- Number fields and threefolds
Reshetikihn, Turaev
software
history
하위페이지
encyclopedia
- http://en.wikipedia.org/wiki/Quantum_invariant
- http://ko.wikipedia.org/wiki/[1]
- http://en.wikipedia.org/wiki/
books
expositions
- Arithmetic properties of quantum invariants of manifolds http://www.mathnet.ru/php/presentation.phtml?presentid=3937&option_lang=rus Don Zagier
- Christian Blanchet, Vladimir Turaev Quantum Invariants of 3-manifolds
articles
- Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links J.M. Borwein, D.J. Broadhurst, 1998
- Gliozzi, F., and R. Tateo. 1995. Thermodynamic Bethe Ansatz and Threefold Triangulations. hep-th/9505102 (May 17). doi:doi:10.1142/S0217751X96001905. http://arxiv.org/abs/hep-th/9505102.
- Three-manifolds and the Temperley-Lieb algebra W. B. R. Lickorish, 1991
- Hyperbolic manifolds and special values of Dedekind zeta-functions Don Zagier, Inventiones Mathematicae, Volume 83, Number 2 / 1986년 6월
question and answers(Math Overflow)
blogs
experts on the field